Ultrasonic Harmonic Generation Research Articles

Systematic investigations on frequency mixing response of ultrasonic shear horizontal and Rayleigh Lamb waves

The study of self- and cross-interactions of shear horizontal (SH) waves has garnered significant attention in the field of nonlinear ultrasonic practice. This interest stems from the potential high sensitivity of SH waves to microdamage and their low out-of-plane attenuation characteristics. However, due to the dispersive nature and intricate mechanisms governing guided wave propagations and interactions, the nonlinear responses of SH waves, particularly the third-order harmonic generation and wave mixing with Rayleigh Lamb (RL) waves, remain largely unexplored. This study presents a comprehensive investigation into the ultrasonic harmonic wave generation of SH guided waves. The research approach combines theoretical considerations and finite element modeling to investigate and discuss guided wave mixing involving SH and RL waves. Six cases of phase-matched wave mode pairs are developed based on deduced synchronism and symmetry conditions. A nonlinear material constitutive model, encompassing both the third- and fourth-order elastic constants, is incorporated into the simulation. The results of the numerical analysis reveal the existence of newly generated higher and combined harmonic waves in an intuitive manner. Specifically, the interactions between ultrasonic SH and RL guided waves are observed to generate second- and third-order combined harmonic waves at the mixing frequencies under internal resonant conditions. The nonlinear waves have a cumulative effect during the mixing period, with their amplitude remaining constant after passing through the mixing zone. This study provides the first observation of mode conversion and third-order combined harmonic generation of SH waves. The insights gained from the numerical perspective offer understandings of the frequency mixing response of SH waves.

Monitoring of stress variation of strands in prestressed concrete by second harmonic generation measurements based on piezoelectric sensors

The stress loss of prestressed concrete may cause the ultimate failure of structures and the monitoring of stress variation of prestressed structures is therefore important to ensure the service safety. This paper developed a nondestructive evaluation method of nonlinear ultrasonic second harmonic generation (SHG) based on piezoelectric (PZT) sensors and applied for the distinction of different stress level of post-tensioned steel strands in concrete. The nonlinear ultrasonic behavior of both free strand and embedded strand having different length is studied with the developed SHG technique and a defined nonlinear parameter is introduced in SHG experiments to correlate with the tension stress in the strand. It is found that the nonlinear parameter has a negative correlation with the increasing stress of strand in both free and embedded case, indicating the nonlinear ultrasonic behavior is getting weak corresponding to the increasing contact force among strand wires. In addition, the decreasing ratio of nonlinear parameter is very close to each other regardless of the length of steel strand, indicating the possibility of replacing the large-scale on-site testing with scaled down samples in laboratory. The experimental results of this research demonstrate that the SHG technique based on PZT sensors could be useful for the stress monitoring of prestressed reinforced concrete with consideration of the relatively high sensitivity of nonlinear parameter to the variation of contact force of strand wires.

Fully noncontact nonlinear ultrasonic characterization of thermal damage in concrete and correlation with microscopic evidence of material cracking

This paper investigates the degradation of concrete integrity under the rising temperature using a nonlinear ultrasonic second harmonic generation (SHG) technique based on a fully non-contact approach. The microscopic cracks accumulated during the course of thermal damage is then analytically and experimentally quantified by the elastic wave velocity inversion method and X-ray computed tomography (CT) technique, respectively. The nonlinear parameter calculated by the non-contact SHG technique not only shows a higher sensitivity to the damage growth than traditional macroscopic ultrasonic parameter such as wave velocity, but also presents an excellent correlation with microscopic quantified crack density by showing a proportionally increase with the propagation distance of ultrasonic waves. The experimental findings in this work indicate that the non-contact SHG technique can truly reveal the progressive microscopic cracking of concrete through the macroscopic nondestructive measurements and is suitable for the assessment of globally distributed damage with a high sensitivity and reliability.

Ultrasonic harmonic generation from materials with up to cubic nonlinearity.

This letter considers the combined effects of quadratic and cubic nonlinearity on plane wave propagation in generally anisotropic elastic solids. Displacement solutions are derived that represent the fundamental, second-, and third-harmonic waves. In arriving at the solutions, the quadratic and cubic nonlinearity parameters for generally anisotropic materials are defined. The effects of quadratic and cubic nonlinearity are shown to influence the amplitude and phase of the fundamental wave. In addition, the phase of the third-harmonic depends on a simple combination of the quadratic and cubic nonlinearity parameters. Nonlinearity parameters are given explicitly for materials having isotropic and cubic symmetry. Lastly, acoustic nonlinearity surfaces are introduced, which illustrate the nonlinearity parameters as a function of various propagation directions in anisotropic materials.

Constitutive model for third harmonic generation in elastic solids

In this article, we present a new constitutive model for studying ultrasonic third harmonic generation in elastic solids. The model is hyperelastic in nature with two parameters characterizing the linear elastic material response and two other parameters characterizing the nonlinear response. The limiting response of the model as the nonlinearity parameters tend to zero is shown to be the well-known St Venant–Kirchhoff model. Also, the symmetric response of the model in tension and compression and its role in third harmonic generation is shown. Numerical simulations are carried out to study third harmonic generation in materials characterized by the proposed constitutive model. Predicted third harmonic guided wave generation reveals an increasing third harmonic content with increasing nonlinearity. On the other hand, the second harmonics are independent of the nonlinearity parameters and are generated due to the geometric nonlinearity. The feasibility of determining the nonlinearity parameters from third harmonic measurements is qualitatively discussed.

On some aspects of material behavior relating microstructure and ultrasonic higher harmonic generation

This article investigates some important aspects of material behavior responsible for acoustic nonlinearity. Even though the discussion is based on a specific constitutive model used for studying higher harmonic generation, the conclusions drawn are valid in a general context. Three aspects of material behavior, namely tension–compression asymmetry, shear-normal coupling and deformation induced anisotropy are presented. The role of each in the generation of higher harmonics along with the plausible microstructural features that contribute to such behavior is discussed. First and foremost, tension–compression asymmetry is identified to cause second (even) harmonic generation in materials. Then shear-normal coupling is identified to cause generation of secondary waves of different polarity than the primary waves. In addition, deformation induced anisotropy due to the presence of residual stress/strain and its contribution to acoustic nonlinearity is qualitatively discussed. Meso-scale modeling aspects to accurately predict the effect of microstructure on higher harmonic generation are emphasized throughout.

Pulse-Echo Harmonic Generation Measurements for Non-destructive Evaluation

Ultrasonic harmonic generation measurements have shown great potential for detecting nonlinear changes in various materials. Despite this, the practical implementation of the technique in non-destructive evaluation (NDE) has typically been limited to the through transmission setup case, with which problems arise in certain situations. Recently, works in the fields of nonlinear fluids and biomedical imaging have reported different application of the harmonic generation theory by making use of reflective boundaries and beam focusing. It is thought that such techniques may be similarly applied in the field of NDE to enable single-sided nonlinear inspection of components. In this paper, we initially describe a numerical model which has been used to determine the effects of attenuation and acoustic beam diffraction on measurements of the nonlinear parameter β. We then extend the model to incorporate first the effects of multiple reflecting boundaries in the propagation medium, then of focused source excitation. Simulations, supported by experimental data, show that nonlinear pulse-echo measurements have the potential to provide a viable (and practical) alternative to the usual through-transmission type as a means of measuring β in solids. Furthermore, it is shown that such measurements may be optimised, both by adjusting the excitation frequency, and by focusing the acoustic source at a certain point relative to the specimen boundary.

Nonlinear acoustics: A transition from laboratory to a practical fatigue assessment tool.

The ultrasonic second harmonic generation technique measuring the nonlinearity parameter has been developed and used in laboratories for several decades throughout the world. Many students and visiting scholars from numerous countries devoted their passions at the physical acoustics laboratories led by Dr. Mack Breazeale for 50 years. During that period of time, nonlinear elastic properties of single crystals, polycrystalline alloys, superconductors, ceramics, and composites were measured. New measurement systems and techniques were also developed to understand temperature dependent linear and nonlinear elastic behaviors of these materials over a temperature range from the liquid helium to Curie temperatures of piezoelectric ceramics. It took years of research and development efforts with a persistent funding before a practical fatigue damage measurement technique was finally developed. The transition from a rack full of equipment to a portable fatigue damage measurement system, a continuous iteration pro...

Ultrasonic harmonic generation from fatigue-induced dislocation substructures in planar slip metals and assessment of remaining fatigue life

An analytical model is presented of the microelastic-plastic nonlinearities resulting from the interactions of a stress perturbation with dislocation substructures and cracks that evolve during cyclic fatigue of planar slip metals. The interactions are quantified by a material nonlinearity parameter β extracted from acoustic (ultrasonic) harmonic generation measurements. The β parameter for a given fatigue state is highly sensitive to the volume fractions of active persistent Luders bands (PLBs) and PLB internal stresses, as well as to the densities, loop lengths, and dipole heights of the dislocation monopoles and dipoles that form the PLBs. The β parameter is predicted to increase monotonically with the increase in the hardness of the metal during cyclic loading, thus allowing an unambiguous assessment of the remaining life of the material. The model is applied to the calculation of β as a function of percent full fatigue life of IN100 nickel-base superalloy. The theoretical predictions are in good agreement with experimental measurements reported in the literature of IN100 samples fatigued in strain-controlled, low cycle, fully reversed loading.

Fatigue damage characterization using surface acoustic wave nonlinearity in aluminum alloy AA7175-T7351

Nonlinear ultrasonic (NLU) harmonic generation system was used to characterize the fatigue damage in a flat hour-glass, high strength Al–Cu–Zn–Mg alloy, AA7175-T7351 specimens. Experiments were carried out to introduce controlled levels of fatigue damage under constant amplitude loading to determine the NLU response using surface acoustic wave (or Rayleigh mode) at regular intervals of fatigue life. The NLU parameter (A2/A12) plotted as a function of percentage of fatigue life shows two peaks for all the samples tested, independent of the amplitude of fatigue loading. The first peak appeared between 40%–50% of fatigue life and the second peak between 80%–90% of fatigue life. Among the two flat surfaces of the specimen, a higher nonlinearity response was observed on the surface which had the first crack initiation. The appearance of two peaks in the nonlinear response during fatigue damage progression is explained based on the dislocation dynamics and dislocation-crack interaction present in the specimens during the fatigue process.

Quantification of fatigue damage accumulation using non-linear ultrasound measurements

This article attempts to relate ultrasonic second harmonic generation to dislocation structures in fatigued materials. By focusing on the fact that asymmetric dislocation motion is required to generate the second harmonic, a theoretical derivation of the non-linearity parameter (ratio of the amplitude of second harmonic to square of the amplitude of the fundamental) from dislocation pile-ups is presented. In order to verify the theory, harmonic generation was measured along a failed fatigue sample of DA718, a nickel base superalloy that exhibits planar slip, and consequently dislocation pile-ups. TEM studies along the length of the failed fatigue sample revealed a banded dislocation structure, composed of pile-ups near the fracture zone and a sparse dislocation network closer to the grip region. The non-linearity parameter was found to increase by between 90% and 140% from the grip region to the fracture zone and this correlated well with the calculations based on theoretical expressions derived here.

Nonlinear properties of lead zirconate–titanate piezoceramics

Nonlinear properties of lead zirconate-titanate (PZT) piezoceramics are investigated using ultrasonic second harmonic generation technique. When a sinusoidal ultrasonic wave of frequency ω0 is sent into a nonlinear material, the second harmonic wave with frequency 2ω0 is generated. Through measuring the absolute amplitude of the fundamental (ω0) and of the second harmonic (2ω0) waves, the ultrasonic nonlinearity parameter β can be determined, which involves certain combinations of the third-order elastic constants and piezoelectric coefficients. We report the measured nonlinear parameters for four types of doped PZT ceramics, PZT-4, PZT-5A, PZT-5H, and PZT-5H HD, which are widely used in practice. We found that the nonlinear parameter is much more sensitive than the linear parameter in responding to microstructural changes in piezoceramics. It could be used to distinguish unpoled samples from depoled samples, which are undistinguishable in terms of the linear parameters.

Determination of precipitate nucleation and growth rates from ultrasonic harmonic generation

An analytical model based on precipitate-dislocation interactions is presented that predicts a decrease in ultrasonic harmonic generation during the nucleation of precipitates and an increase during precipitate growth up to the point of precipitate-matrix coherency loss. The opposing effects allow independent, indirect assessments of nucleation and growth rates from a curve fit of the model equation to experimental data. Application to the assessment of S′ precipitation during the artificial aging of AA2024 yields a constant nucleation rate of 5.4×1015 m−3 s−1 and a growth rate increasing as the 1.2 power of time in the range 72–432 min during heat treatment at 463 K.

Ultrasonic Linear and Nonlinear Behavior of Fatigued Ti–6Al–4V

The change in ultrasonic nonlinear property of a titanium alloy subjected to cyclic loading has been studied, with an objective to develop a new characterization methodology for quantifying the level of damage in the material undergoing fatigue. In order to determine the degree of nonlinearity, the ultrasonic second harmonic generation technique has been used. The second harmonic signal was monitored during the fatigue process, and a substantial increase in the second harmonic amplitude (180% increase in nonlinear factor) was observed. This indicates that the second harmonic signal is very sensitive to the microstructural changes in the material caused by fatigue.

Stimulated surface crystallization of β–barium borate on glass due to ultrasonic treatment and second harmonic generation

A glass of 47.5BaO · 47.5B2O3 · 5Al2O3 (in mol%) having a shorter optical absorption edge (∼236 nm) and suitable glass formation was selected as the mother glass for studying the surface crystallization of β−BaB2O4 · β−BaB2O4 was not the main surface crystallized phase when only a conventional heat treatment was applied. Crystallization of β−BaB2O4 was stimulated due to ultrasonic surface treatment with an ethanol suspension of β−BaB2O4 particles and subsequent heat treatment. After the ultrasonic treatment, β−BaB2O4 was the main crystallized phase. Transparent and dense β−BaB2O4 thin films/glass showed second harmonic generation. The tensor components of the second-order nonlinear optical susceptibility were estimated.

Ultrasonic second harmonic generation in various crystalline systems. II. Piezoelectric materials: Coupling parameters in terms of elastic and piezoelectric moduli and propagation directions

Expressions for the piezoelectric corrections to the second- and third-order coupling parameters K2 and K3 in cubic, rhombohedral, tetragonal, and orthorhombic crystalline structures are presented. The corrections depend on the linear and nonlinear dielectric constants, second- and third-order piezoelectric moduli (which are related to the elasto-optic, electrostrictive, and the electroacoustic coefficients). The corrections are estimated for the cases of crystalline quartz and lithium niobate.

Elastic nonlinearity parameters of tetragonal crystals

The equations of motion for the propagation of finite amplitude elastic waves in crystals of tetragonal symmetry have been derived starting from the expression for the elastic strain energy. The equations have been solved for a finite amplitude sinusoidal wave propagating along the pure mode directions which are [100], [110] and [001] for the tetragonal group TI. The solutions corresponding to longitudinal wave propagation yield expressions for the amplitudes of the fundamental and generated second harmonic for these directions in terms of certain combinations of second and third order elastic constants of the medium. The results will aid the experimenter to determine these constants using ultrasonic harmonic generation technique.

Temperature dependence of the ultrasonic nonlinearity parameters of NaCl

The ultrasonic harmonic generation technique using a capacitive detector has been used to measure the ultrasonic nonlinearity parameters of sodium chloride single crystals between 300 and 77 K. Since NaCl is a typical ionic crystal, its ordinary elastic constants have been of interest to many physicists. Likewise, its nonlinear properties, its third‐order elastic constants, and their temperature dependence have been of interest to solid‐state theory. However, to date the large thermal expansion coefficient and the fragility of NaCl have thwarted measurement of the temperature dependence of its nonlinearity parameters. Differential expansion between NaCl and a bonded quartz transducer often has resulted in sample cleavage. A special technique for bonding the quartz transducer to the NaCl sample has been developed. With the aid of this technique, ultrasonic nonlinearity parameters of three different oriented samples of NaCl have been measured for the first time in the temperature ranges of 300 to 77 K for the (100) sample, 300 to 100 K for the (110) sample, and 300 to 110 K for the (111) sample. The TOE constant C111 and two combinations of the other TOE constants and their temperature dependence have been obtained. The results are compared with theory describing interatomic forces in alkali halide crystals. (Research supported in part by the ONR and the UT‐ORNL Science Alliance.]

Relationship between the stress acoustic constants and the ultrasonic nonlinearity parameters for solids

The ultrasonic characterization of the stress state of a solid is usually based on the measurement of the stress acoustic constant evaluated at the stress free state for each material under investigation. For isotropic and quasi-isotropic solids and for crystals along nonrotated pure mode propagation directions the stress acoustic constant can be obtained from ultrasonic harmonic generation measurements. The relationships among the acoustic constant, the strain acoustic constant, and the ultrasonic nonlinearity parameters for solids will be derived. Experimental results verifying these relationships will be presented for a number of materials.

Third-order elastic constants and Grüneisen parameters of silicon and germanium between 3 and 300 °K

We present a method to determine all six independent third-order elastic (TOE) constants of diamond-like solids as a function of temperature from measured values of the ultrasonic nonlinearity parameters. Ultrasonic harmonic generation experiments along the three principal directions of a cubic crystal lead to the determination of the nonlinearity parameters, which contain three linear combinations of TOE constants. These data are sufficient to determine the three anharmonic first and second neighbor force constants in the Keating model for diamond-like solids. Ultrasonic velocity data are used to determine the two harmonic force constants in this model. Since our experiments have given data on silicon and germanium between liquid helium temperature and room temperature, we are able to evaluate and plot the Keating model force constants and hence all the six third-order elastic constants between 3 and 300 °K. Most of the TOE constants of silicon and germanium are smoothly varying functions of temperature; however, C123 and C144 go through zero and assume positive values at low temperatures. To determine whether the behavior of the calculated TOE constants is reasonable, we have used them to calculate the generalized Grüneisen parameters γ in the quasiharmonic approximation and have plotted γ as a function of temperature. These curves are in better agreement with curves obtained from thermal expansion experiments than the curves plotted by earlier workers using room temperature values of third-order elastic constants. For completeness we also have calculated the pressure dependence of the SOE constants and the Anderson Grüneisen parameters over the same temperature range.